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Theorem rankelop 8292
Description: Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)
Hypotheses
Ref Expression
rankelun.1  |-  A  e. 
_V
rankelun.2  |-  B  e. 
_V
rankelun.3  |-  C  e. 
_V
rankelun.4  |-  D  e. 
_V
Assertion
Ref Expression
rankelop  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  <. A ,  B >. )  e.  ( rank `  <. C ,  D >. ) )

Proof of Theorem rankelop
StepHypRef Expression
1 rankelun.1 . . . 4  |-  A  e. 
_V
2 rankelun.2 . . . 4  |-  B  e. 
_V
3 rankelun.3 . . . 4  |-  C  e. 
_V
4 rankelun.4 . . . 4  |-  D  e. 
_V
51, 2, 3, 4rankelpr 8291 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } ) )
6 rankon 8213 . . . . 5  |-  ( rank `  { C ,  D } )  e.  On
76onordi 4982 . . . 4  |-  Ord  ( rank `  { C ,  D } )
8 ordsucelsuc 6641 . . . 4  |-  ( Ord  ( rank `  { C ,  D }
)  ->  ( ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } )  <->  suc  ( rank `  { A ,  B } )  e.  suc  ( rank `  { C ,  D } ) ) )
97, 8ax-mp 5 . . 3  |-  ( (
rank `  { A ,  B } )  e.  ( rank `  { C ,  D }
)  <->  suc  ( rank `  { A ,  B }
)  e.  suc  ( rank `  { C ,  D } ) )
105, 9sylib 196 . 2  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  suc  ( rank `  { A ,  B } )  e. 
suc  ( rank `  { C ,  D }
) )
111, 2rankop 8276 . . 3  |-  ( rank `  <. A ,  B >. )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
121, 2rankpr 8275 . . . 4  |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
13 suceq 4943 . . . 4  |-  ( (
rank `  { A ,  B } )  =  suc  ( ( rank `  A )  u.  ( rank `  B ) )  ->  suc  ( rank `  { A ,  B } )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) ) )
1412, 13ax-mp 5 . . 3  |-  suc  ( rank `  { A ,  B } )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
1511, 14eqtr4i 2499 . 2  |-  ( rank `  <. A ,  B >. )  =  suc  ( rank `  { A ,  B } )
163, 4rankop 8276 . . 3  |-  ( rank `  <. C ,  D >. )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
173, 4rankpr 8275 . . . 4  |-  ( rank `  { C ,  D } )  =  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
18 suceq 4943 . . . 4  |-  ( (
rank `  { C ,  D } )  =  suc  ( ( rank `  C )  u.  ( rank `  D ) )  ->  suc  ( rank `  { C ,  D } )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) ) )
1917, 18ax-mp 5 . . 3  |-  suc  ( rank `  { C ,  D } )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
2016, 19eqtr4i 2499 . 2  |-  ( rank `  <. C ,  D >. )  =  suc  ( rank `  { C ,  D } )
2110, 15, 203eltr4g 2573 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  <. A ,  B >. )  e.  ( rank `  <. C ,  D >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474   {cpr 4029   <.cop 4033   Ord word 4877   suc csuc 4880   ` cfv 5588   rankcrnk 8181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-recs 7042  df-rdg 7076  df-r1 8182  df-rank 8183
This theorem is referenced by: (None)
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